- #31
mathdad
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Ok. Great. Let us move on. I will post similar word problems in the coming days.
Click For Summary
Discussion Overview
The discussion revolves around the motion of two balls thrown vertically upward from different heights and with different initial speeds. Participants explore the use of a quadratic formula to determine the time it takes for each ball to hit the ground, as well as the implications of their respective speeds and heights on this outcome. The conversation includes aspects of mathematical reasoning and conceptual clarification related to kinematics.
Discussion Character
- Mathematical reasoning, Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants propose using the formula h(t) = -16t^2 + v_0t + h_0 to model the height of the balls over time.
- There is a discussion about whether to apply the formula separately for each ball or to derive a single formula that can be used for both.
- One participant suggests using the quadratic formula to solve for t, specifically t = (v_0 + sqrt(v_0^2 + 64h_0)) / 32, and notes the importance of discarding negative roots.
- Another participant questions the origin of the constants 64 and 32 in the derived formula.
- Some participants clarify that the ball that hits the ground first is determined by the time taken, not necessarily by the speed at which it hits the ground.
- There is a mention of the limitations of the model, including neglecting air resistance and the effects of gravity on other celestial bodies.
- Participants discuss the implications of throwing objects downward and how the initial velocity would be represented in the formula.
- There is a consideration of how drag affects motion and its relation to differential equations in physics.
Areas of Agreement / Disagreement
Participants generally agree on the use of the quadratic formula for solving the problem, but there are differing views on the interpretation of speed versus time to impact, as well as the application of the model to different scenarios involving vertical motion.
Contextual Notes
The discussion highlights the need for careful consideration of initial conditions and the assumptions underlying the mathematical models used. There is also an acknowledgment of the complexity introduced by factors such as air resistance and varying gravitational forces.
Who May Find This Useful
This discussion may be useful for students and enthusiasts of physics and mathematics who are interested in kinematics, particularly in understanding the motion of objects under the influence of gravity.
- #32
MarkFL
Gold Member
MHB
- 13,284
- 12
Okay, so we've found:
$$t=\frac{v_0\pm\sqrt{v_0^2-2a\Delta x}}{a}$$
Suppose we have $\Delta x=0$. At what times does that occur?
$$t=\frac{v_0\pm\sqrt{v_0^2-2a\Delta x}}{a}$$
Suppose we have $\Delta x=0$. At what times does that occur?
- #33
mathdad
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MarkFL said:
Replace (delta)x with 0 and simplify, right?
- #34
MarkFL
Gold Member
MHB
- 13,284
- 12
RTCNTC said:
Yes. :)
- #35
mathdad
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Thank you for all your help.
- #36
MarkFL
Gold Member
MHB
- 13,284
- 12
Um, okay.
I was attempting to share with you some of the ways I used to go about learning things that helped me succeed in my math studies, but if you just wish to walk away from that, then so be it. Examining formulas at the boundaries of the parameter values gave me a good number of "aha" moments. Looking at how one values changes in relation to another is also illuminating.
I was attempting to share with you some of the ways I used to go about learning things that helped me succeed in my math studies, but if you just wish to walk away from that, then so be it. Examining formulas at the boundaries of the parameter values gave me a good number of "aha" moments. Looking at how one values changes in relation to another is also illuminating.
- #37
mathdad
- 1,280
- 0
MarkFL said:
I thank you for helping me as I march on my way to succeed with word problems.
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